This month's topics:

Alexander Grothendieck, 1928-2014

Alexander Grothendieck died on November 13, 2014. His obituary in the next day's New York Times was titled: "Alexander Grothendieck, Math Enigma, Dies at 86," referring probably both to the bewildering power of his mathematical genius and to the mystery with which he cloaked his last years, living in seclusion in a village in the Pyrenees; he died nearby. The Times writers, Bruce Weber and Julie Rehmeyer, set the stage for their account of Grothendieck's work: "Algebraic geometry is a field of pure mathematics that studies the relationships between equations and geometric spaces. Mr. Grothendieck was able to answer concrete questions about these relationships by finding universal mathematical principles that could shed unexpected light on them." They come back to this organizing principle of Grothendieck's mathematics later in the obituary, speaking of his contribution to the proof of the Weil Conjectures: "But characteristically he did not attack the problem directly. Instead, he built a superstructure of theory around the problem. The solution then emerged easily and naturally, in a way that made mathematicians see how the conjectures had to be true. He avoided clever tricks that proved the theorem but did not develop insight. He likened his approach to softening a walnut in water so that, as he wrote, it can be peeled open 'like a perfectly ripened avocado.'" And they include this quotation from his memoir Reapings and Sowings:

"If there is one thing in mathematics that fascinates me more than anything else (and doubtless always has), it is neither 'number' nor 'size,' but always form. And among the thousand-and-one faces whereby form chooses to reveal itself to us, the one that fascinates me more than any other and continues to fascinate me, is the structure hidden in mathematical things."

The Times also published, on November 25, "The Lives of Alexander Grothendieck, a Mathematical Visionary" by Edward Frenkel. The essay devotes equal space to Grothendieck's mathematical accomplishments (Frenkel actually gives an example of what algebraic geometry is about) and to his equally obsessive work on human rights and environmental degradation. Frenkel connects the two lives: "Though one might ask if there are any real-world applications of his work, the more important question is whether having found applications, we also find the wisdom to protect the world from the monsters we create using these applications. Alas, the recent misuse of mathematics does not give us much comfort." He links to the page Grothendieck's non-mathematical writings, which itself links to issues of the newsletter Survivre et Vivre, published in 1970-73. There, Frenkel tells us, "one can see Grothendieck confronting the world's ills with his signature rigor and passion."

Le Monde ran the headline, on November 14, "Alexandre Grothendieck, the greatest mathematician of the XX century, has died." Their obituary (by Stéphane Foucart and Philippe Pajot) contains many biographical details, including the legend of the fourteen problems:

"Looking for a thesis problem, he is sent to meet Laurent Schwartz and Jean Dieudonné. ... The two great mathematicians give the young student a list of fourteen problems which they consider a vast program of research for the years to come, and ask him to choose one. A few months later, Alexandre Grothendieck is back: he has solved them all."

And this quote from his student Pierre Deligne: "He was unique in his way of thinking. He had to understand things from the most general point of view possible; once things had been settled and understood in that way, the landscape would become so clear that proofs seemed almost trivial." Le Monde online has a six-minute video in which the mathematician and historian Jean-Michel Kantor gives an eloquent portrayal of Grothendieck and his scientific impact, including a reading from Grothendieck's text of the entire walnut-avocado simile mentioned in the Times. The title of the video, "Grothendieck's ideas have penetrated the subconscious of mathematicians" is a quote from Pierre Deligne. Le Monde also gives a link to the text of the letter Grothendieck sent them in 1988, explaining his refusal of the Crafoord Prize.

Also available on the web is A country known only by name, written by his former associate Pierre Cartier: a detailed overview of Grothendieck's scientific work, along with a first-person account of some of the stormier moments that punctuated his withdrawal from academic and scientific life. [My translations, except for those quoted; Cartier's text has been translated into English, but Reapings and Sowings, as far as I know, unfortunately, has not. -TP]

New Zealand robin arithmetic

An article in press in Behavioural Processes was picked up by the website Nature World News ("Birds Can Count: How We Know It," November 18, 2014), on the conservation website The Dodo (Birds Literally Do The Math When Their Mate's Behavior Doesn't Add Up) and featured in a video on the "Science Take" webpage of the New York Times (by David Frank, November 17, 2014). The article is "Addition and Subtraction in wild New Zealand Robins," by Alexis Garland and Jason Low (Victoria University, New Zealand). Garland and Low presented robins in the wild with a "Violation of Expectancy" (VoE) task designed to test their discrimination of number, and changes in number. The test used a box with two compartments, similar in design to those used in disappearing-penny "magic" tricks; instead of penny/no-penny, different numbers of meal-worms were placed in the compartments, as follows: . "... the upper compartment of the VoE box ... was first baited out of view (pre-trial) with the final quantity of prey found by the robin, then the trial was initiated, and a quantity of prey added (and in some cases subtracted) from the apparatus within view of the robin ..." Note within view of the robin. The worms go into or are taken from the lower compartment. Then a leather disc is placed over the box and the upper compartment is secretly slid in over the lower. " ... finally the experimenter stepped back, and the robin was allowed to uncover the apparatus and access the (now visible) upper compartment." "Robins spend the majority of their time hunting on the forest floor, turning over leaves in search of insects ... . As such, pulling the leather flap from a small wooden platform was a very simple extension of their natural behaviour, adopted typically within a very short period of exposure to the materials (well under 30 min, on average)."

"Robins were shown 8 different hiding events in randomised order, and found 4 numerically congruent and 4 numerically incongruent" as itemized in this table:

Congruent

Incongruent

$1+0=1$

$1+1\neq 1$

$1+1=2$

$2-0\neq 1$

$2-1=1$

$3-0\neq 2$

$3-1=2$

$3-1\neq 1$

The authors measured robins' activity in the period after the flap was lifted. "A video analysis was performed looking at 2 different dimensions of response behaviour: First, search duration -- the total amount of time the robin spent actively examining the apparatus (looking closely into, at or under it, pecking at it, hopping onto or around it) or leather cover (pulling at it with their beak, flipping it over, standing on it, looking closely at it). Second, pecking frequency -- the number of times the subject pecked with its beak at any part of the apparatus."

The results of the described experiment: average response for search time(s) and for number of pecks compared between the "congruent" trials (green; see table above) and the "incongruent" trials (orange). Error bars show $\pm 1$ Standard Error. Image adapted from Garland and Low.

As the authors report: "... on average, robins measured higher on both behavioural measures in incongruent than congruent trials." And they conclude: "... robins appear to be able to respond to proto-numerical summation and subtraction involving small ($<4$) quantities." As to where mates come into the picture, "In addition to hunting and caching insects, pairs also frequently pilfer prey from mates."

"Solving for XX:" women and math in the Berkeley Daily Planet

Jonathan Farley contributed the Feature "Solving for XX" to the Berkeley Daily Planet for October 23, 2014. The highlight is an interview with Danica McKellar, the actress with the math degree from UCLA and the author of, inter alia,Kiss My Math: showing pre-algebra who's boss. McKellar: "The problem isn't that girls don't do math as well as boys. The problem is that, in spite of good test scores, girls don't see themselves as capable of doing math as well as boys. So as soon as they hit a stumbling block, instead of seeing it as a temporary obstacle that can be overcome, they more often see it as evidence of what they've 'known' all along -- that they don't belong in math. That it's not really 'for them.' ... the only way around it is to do what we can to break stereotypes, and to bombard girls with the opposite of the limiting female characters they get from most media: positive role models to show them, 'You have every potential within you. Develop your brain. You belong!'"

Knots in physics

"Get Knotted: They've been practising for ages, but physicists are finally learning how to tie knots in things," by Leonie Mueck, ran in the New Scientist on October 4, 2014. The short article surveys physicists' involvement with knots, starting back in the 19th century with the idea, due to William Thomson (later named Lord Kelvin), that atoms might correspond to infinitesimal looped vortices in the "lumeniferous
aether," with different elements corresponding to different knots.
This led to Thomson's collaborator
Peter Tait drawing up the first knot tables, but was a dead end in physics. Mueck gives a quick survey of recent developments, which include: